In the pattern of fundamental UIR's for the four classical families, we found an unusual feature for the real orthogonal groups Dl = SO (2l) and Bl = SO (2l + 1). The fundamental UIR's are not all obtainable from the defining ‘vector’ representation D by tensorial constructions. At the level of elementary UIR's this is seen even more clearly. For Al = SU (l + 1) and Cl = USp (2l) , we have only one elementary UIR in each case, namely D itself. But for both Bl and Dl , the elementary UIR's consist of the ‘vector’ UIR D , and either one or two spinor UIR's (These are actually double valued UIR's of the respective groups SO (2l + 1), SO (2l) .) We study them briefly in this chapter.

Cartan had found the spinor UIR's by around 1913. Independently, Dirac found them in 1928 for the Lorentz group, and then they entered physics. Of course even earlier spinors for SU (2) and SO (3) were used in Pauli's description of spin in the non relativistic framework. In 1935 Brauer andWeyl [Brauer andWeyl, 1935] gave an elegant account of Cartan's fundamental spinor UIR's for Bl and Dl using the Dirac approach, i.e., via the algebra of γ matrices which we develop below.

Spinor UIR's forDl =SO (2l)

We see from the discussion in Section 5.3 that the defining vector representation D of SO (2l) of dimension 2l , has the following highest weight and other weights making

All these weights are simple. It is clear that in any UIR formed out of tensors over D , all weights present will be (positive or negative) integer linear combinations of the ea.